Answered: 1. An incompressible Newtonian liquid…

Introduction to Chemical Engineering Thermodynamics

8th Edition

ISBN:9781259696527

Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Chapter1: Introduction

Section: Chapter Questions

Problem 1.1P

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$## Problem 1: Fluid Dynamics in a Slit Consider an incompressible Newtonian liquid in a steady laminar flow through a slit formed by two parallel vertical walls separated by a distance of 2B. The walls have a length L and a width W. Given that B is much smaller than both W and L, edge and end effects are negligible. The flow is fully developed and varies only in the x-direction. The pressure is $ P_o $ at the entrance and $ P_L $ at the exit. ### Diagram Overview - **Diagram Description**: The diagram depicts two parallel walls with length L, width W, and separation 2B. Fluid enters from the top (Fluid in) and exits at the bottom (Fluid out), moving along the length L. The coordinate system is centered such that $ x = 0 $ is midway between the two vertical surfaces, and the fluid velocity varies along the z-axis. ### Tasks (a) Write the simplified form of the Navier-Stokes equation used to solve for the velocity profile $ v_z $. (b) Derive the two boundary conditions necessary for obtaining a solution for the velocity profile. (c) Obtain a solution for the velocity profile $ v_z $ as a function of x, involving parameters $ P_o, P_L, \rho, g, \mu, L, $ and $ B $. (d) Determine the maximum velocity and its location within the flow. (e) Derive equations for the volumetric flow rate and average velocity. Explore the relationship between the average and maximum velocity.$

Transcribed Image Text:## Problem 1: Fluid Dynamics in a Slit Consider an incompressible Newtonian liquid in a steady laminar flow through a slit formed by two parallel vertical walls separated by a distance of 2B. The walls have a length L and a width W. Given that B is much smaller than both W and L, edge and end effects are negligible. The flow is fully developed and varies only in the x-direction. The pressure is $ P_o $ at the entrance and $ P_L $ at the exit. ### Diagram Overview - **Diagram Description**: The diagram depicts two parallel walls with length L, width W, and separation 2B. Fluid enters from the top (Fluid in) and exits at the bottom (Fluid out), moving along the length L. The coordinate system is centered such that $ x = 0 $ is midway between the two vertical surfaces, and the fluid velocity varies along the z-axis. ### Tasks (a) Write the simplified form of the Navier-Stokes equation used to solve for the velocity profile $ v_z $. (b) Derive the two boundary conditions necessary for obtaining a solution for the velocity profile. (c) Obtain a solution for the velocity profile $ v_z $ as a function of x, involving parameters $ P_o, P_L, \rho, g, \mu, L, $ and $ B $. (d) Determine the maximum velocity and its location within the flow. (e) Derive equations for the volumetric flow rate and average velocity. Explore the relationship between the average and maximum velocity.

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